Objective:
To show that the figure obtained by joining the mid-points of consecutive sides of the quadrilateral is a parallelogram.
Parallelogram:
A parallelogram is a simple quadrilateral with two pairs of parallel sides.
The opposite or facing sides of a parallelogram are of equal length.
Proof:
In quadrilateral ABCD points P, Q, R, S are midpoints of side AB, BC, CD and AD respectively.
To prove :
PS || QR and SR || PQ. i.e. Quadrilateral PQRS is a parallelogram
Proof:
Draw diagonal BD.
As PS is the midsegment of ▲ ABD, we can say that PS || BD.
As QR is the midsegment of ▲ BCD, we can say that QR || BD.
∵ PS || BD and QR || BD by transitivity, we can say that PS || QR.
Now draw diagonal AC.
As SR is the midsegment of ▲ ACD, we can say that SR || AC.
As PQ is the midsegment of ▲ ABC, we can say that PQ || AC.
∵ SR || AC and PQ || AC by transitivity, we can say that SR || PQ.
∵ PS || QR and SR || PQ, ∴ quadrilateral PQRS is a parallelogram (by definition).